Automated searching for quantum subsystem codes

Quantum error correction allows for faulty quantum systems to behave in an effectively error-free manner. One important class of techniques for quantum error correction is the class of quantum subsystem codes, which are relevant both to active quantum error-correcting schemes as well as to the design of self-correcting quantum memories. Previous approaches for investigating these codes have focused on applying theoretical analysis to look for interesting codes and to investigate their properties. In this paper we present an alternative approach that uses computational analysis to accomplish the same goals. Specifically, we present an algorithm that computes the optimal quantum subsystem code that can be implemented given an arbitrary set of measurement operators that are tensor products of Pauli operators. We then demonstrate the utility of this algorithm by performing a systematic investigation of the quantum subsystem codes that exist in the setting where the interactions are limited to two-body interactions between neighbors on lattices derived from the convex uniform tilings of the plane.

Figure 1

Illustration of two possible labelings of a graph, which correspond to two possible choices of two-body measurements. In both graphs we see that there are four vertices and five edges, which indicates that our system is constrained to have four qubits and five two-body measurement operators. The edges (without the labels) indicate the pairs of qubits on which the two-body measurement operators are constrained to act within our system. Within these constraints, we see in this figure two possible choices of measurement operators as specified by the two labelings of the edges of the graph.

Figure 2

Illustration of how we placed the centers and boundaries for each of the tilings that we scanned. The boundaries are periodic, so that edges that pass through one side of the boundary wrap around to the opposite side. Edges and vertices $\mathrm{not}$ a boundary are merged with the corresponding edges and vertices on the opposite edge. Note that under this scheme there can be no vertices on a corner. For most tilings this will never happen, but it turns out that in the case of the deltille tiling there are vertices on the corners when the radius (smallest distance from the center to the boundary) is three times the radius of the unit cell; we thus ignore deltille tilings of these sizes.

Figure 3

Plots of the results from scanning the (a) quadrille, (b) isosnub quadrille, (c) truncated quadrille, and (d) snub quadrille tilings. Every polygon in the plot corresponds to a code that was found with a distance equal to the number of sides of the polygon, so that triangles indicate distance-3 codes, squares indicate distance-4 codes, etc. The position along the $x$ axis indicates the radius of the lattice where the code was found, where the radius is an integer defined to be the length of the lattice divided by the length of the smallest possible periodic lattice for the tiling; it also indicates the number of physical qubits in the lattice where the code was found, which appears on the $x$ axis just under the value of the radius. The position along the $y$ axis indicates the number of logical qubits with that distance in the code. Note that in cases where multiple codes were found for the same radius and with the same number of logical qubits, multiple polygons are drawn, so that, for example, in plot (c) we see several cases in which a distance-3 code (triangle) and a distance-4 code (square) were found that were in lattices with the same radius and also had the same number of logical qubits.

Figure 4

Plots of the results from scanning the (a) hextille, (b) rhombihexadeltille, (c) truncated hextille, and (d) hexadeltille tilings. See the caption of Fig. 3 for an explanation of how to interpret these plots.

Figure 5

Illustration of the single labeling of the quadrille tiling (on a radius-2 lattice) that results in a useful quantum code.

Figure 6

Illustration of four labelings of the truncated quadrille tiling (on a radius-1 lattice) that result in a quantum code with distance 3 (4) [in respectively (a) and (b)], and a number of logical qubits proportional to the square of the radius of the labeling.

Figure 7

Illustration of a labeling of the snub quadrille tiling (on a radius-2 lattice) that results with distance 4 and a number of distance logical qubits proportional to the square of the radius of the labeling.

Figure 8

Illustration of two of the labelings of the snub quadrille tiling (on a radius-2 lattice) that result in a quantum code with a single qubit whose distance grows with the radius of the lattice.

Figure 9

Illustration of two of the labelings of the isosnub quadrille tiling (on a radius-1 lattice) that result in a quantum code with a single qubit whose distance grows with the radius of the lattice.

Figure 10

Illustration of four labelings of the hextille tiling (on a radius-2 lattice) that result in a quantum code with distance 3 (4) [in respectively (a) and (b)], and a number of distance logical qubits proportional to the square of the radius of the labeling.

Figure 11

Illustration of two labelings of the truncated hextille tiling (on a radius-1 lattice) that result in a quantum code with distance 3 and a number of distance logical qubits proportional to the square of the radius of the labeling.

Figure 12

Illustration of two labelings of the truncated hextille tiling (on a radius-1 lattice) that result in a quantum code with distance 4 and a number of distance logical qubits proportional to the square of the radius of the labeling.

Figure 13

Illustration of one of the labelings of the hexadeltille tiling (on a radius-1 lattice) that results in quantum code whose distance grows with the size of the tiling.

Figure 14

Illustration of labelings of the rhombihexadeltille tiling (on a radius-2 lattice) that result in a quantum code with distance 5 and four qubits (a) and a quantum code with distance 6 and two qubits (b) when applied to a radius-3 lattice.

Figure 15

Illustration of two labelings of the rhombihexadeltille tiling (on a radius-2 lattice) that result in a quantum code with distance 4 and 16 qubits when applied to a radius-3 lattice.